Consider a real-valued function \(f:\overline{{\cal M}}_{\mathbf {r}}/{\cal G}\rightarrow \mathbb {R}\)  and its lift \(\bar{f} = f\circ \pi : \overline{{\cal M}}_{\mathbf {r}}\rightarrow \mathbb {R}\) .
By [1]},
the horizontal lift of the Riemannian gradient of \(f\)  can be obtained from the Riemannian gradient of \(\bar{f}\) :
\(\overline{\operatorname{grad}f\left(\pi \left(\bar{{\cal X}}\right)\right)} = \operatorname{grad}\bar{f}\left(\bar{{\cal X}}\right).\) 
